www.biomathforum.org/biomath/index.php/biomath ORIGINAL ARTICLE COVID-19 changing the face of the world. Can sub-Sahara Africa cope? Mandidayingeyi H Machingauta1,2, Bwalya Lungu3, Edward M Lungu1 1 Department of Mathematics and Statistical Sciences Botswana International University of Science and Technology Palapye, Botswana hellen.machingauta@studentmail.biust.ac.bw, lungue@biust.ac.bw 2Department of Surveying and Geomatics Midlands State University, Gweru, Zimbabwe pfupajenamh@staff.msu.ac.zw 3Department of Food Science University of California,Davis, CA 95616, USA blungu@ucdavis.edu Received: 7 September 2020, accepted: 11 March 2021, published: 29 March 2021 Abstract— We formulate a mathematical model for the spread of the coronavirus which incorporates adherence to disease prevention. The major results of this study are: first, we determined optimal infec- tion coefficients such that high levels of coronavirus transmission are prevented. Secondly, we have found that there exists several optimal pairs of removal rates, from the general population of asymptomatic and symptomatic infectives respectively that can protect hospital bed capacity and flatten the hospital admission curve. Of the many optimal strategies, this study recommends the pair that yields the least number of coronavirus related deaths. The results for South Africa, which is better placed than the other sub-Sahara African countries, show that failure to address hygiene and adherence issues will preclude the existence of an optimal strategy and could result in a more severe epidemic than the Italian COVID-19 epidemic. Relaxing lockdown measures to allow individuals to attend to vital needs such as food replenishment increases household and community infection rates and the severity of the overall infection. Keywords-COVID-19; Hospital bed capacity; Re- moval rates; Optimal strategies I. INTRODUCTION The coronavirus pandemic has disrupted global economies and health systems in unprecedented ways. As of 20 January 2021 there were 96 715 656 recorded cases with 2 068 062 deaths Copyright: © 2021 Machingauta et al. This article is distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Citation: Mandidayingeyi H Machingauta, Bwalya Lungu, Edward M Lungu, COVID-19 changing the face of the world. Can sub-Sahara Africa cope?, Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 1 of 23 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... and 69 410 634 recoveries, [34]. The coronavirus currently infecting humans for the first time started in China in November 2019 in the city of Wuhan. The virus is being referred to as SARS-CoV-2 to differentiate it from other coronaviruses, [33]. Ac- cording to the Centers for Disease Control (CDC) the virus displays symptoms within 2 − 14 days of exposure that may include fever, body aches, dry cough, fatigue, chills, headache, sputum pro- duction, myalgias, anorexia, sore throat, shortness of breath, loss of appetite and loss of smell. Most of these symptoms are similar to the symptoms of diseases caused by other pathogens such as bacteria (e.g tuberculosis, cholera etc), viruses (e.g influenza, HIV, SARS, MERS) and parasites (e.g malaria). In its severe form COVID-19 induces severe pneumonia and has led to high rates of death in about 1 − 5% of those infected [31]. However, some people infected with the virus may show little to no symptoms and are classified as asymptomatic carriers that are still able to transmit the virus to other individuals. These individuals play a major role in transmission as they are silent spreaders and it is challenging to determine the rate at which they spread the disease. There are currently no vaccines or treatments available for COVID-19, but patients suffering from severe symptoms are usually hospitalized with median hospital stays of 10 to 13 days [9], [31], [37]. Most of the countries experiencing high numbers of COVID-19 cases are developed countries including the United States of Amer- ica (USA), Italy, Spain, United Kingdom (UK), France, Germany, Russia, Brazil and China. Most of these developed countries have excellent health facilities. However, the COVID-19 epidemic in Italy, Spain, France and the USA has demonstrated that the current medical facilities were not de- signed to serve the populations during a pandemic. Moreover, the democratic system of governance which we have all cherished has been seriously challenged during these times when some form of authoritarian governance was required in order to enforce prevention measures such as social distancing, masks, lockdown etc [6], [21], [27], to control virus transmission. The current Western health systems performed well during the initial stages of the coronavirus disease progression but have since been challenged due to the acute rise in infection rates. Various countries have had to make decisions over who is offered or not offered a bed in an Intensive Care Unit (ICU). The decisions were based on hospital bed capacity to avoid hospital overload and were dependent on early testing and isolating those who test positive as was the case in South Korea, Germany and China [16]. The epidemic in these developed countries has had very serious effects on the overall infrastruc- ture and livelihoods of the people living there, and we believe that the effects on the sub-Sahara Africa region could very likely overwhelm this region where the economies are very weak and their ICU capacity compared to population sizes averages only 9 beds per 10 000 inhabitants [32]. The current economic landscape will likely impact the ability of the various health systems in the long term to supply personal protective equipment (PPE) required by the frontline workers as they take care of infected patients. The main mode of transmission of the virus is through person to person transmission and this can happen through droplets, airborne transmission, surface transmission and the fecal oral route. The required measures to stop the spread of COVID-19 include social distancing, face mask requirements for out of home activities, staying home if one feels sick or is nursing a cold. Good hygiene practices such as frequent washing of hands or not touching the face area especially the mouth, nose and eyes, good sanitary practices that include frequent sanitation of surfaces. Drastic measures have included mandatory shelter-ins or lockdowns. While these types of measures are not new and have been shown to work, they will likely present a different set of problems for sub-Sahara Africa. The problems sub-Sahara Africa will face will be compounded but not limited to the following: The low standards of good sanitary and hy- giene measures may present an environment that Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 2 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... supports virus spread. Good sanitary and hygiene practices have been encouraged and used in previ- ous epidemics for thousands of years. The avail- ability of clean water and poor hygiene (sanitary) will likely fuel the sub-Sahara coronavirus epi- demic as has been the case for other ongoing epidemics such as the Ebola and cholera. The Ebola [17] and cholera epidemics which remain unresolved will compete with the coronavirus for health facilities and workers. The current health facilities are already inadequate, for example the number of hospital beds 9 per 10 000 people [32], will prove inadequate based on the higher hospitalization rates reported in Italy, France and Spain for individuals infected with the coronavirus [24]. The average family in the poorest country in sub-Sahara Africa lives on less than $1 per day and most of what they earn goes towards food and other necessities. Most families will not be able to afford the cost of soap and sanitizers to maintain the high levels of hygiene required to control or stop the spread of COVID-19 [26]. As a result of inadequate water supplies and lack of sanitizers/soaps a large number of the population may not practice good sanitation and hygiene prac- tices. In addition, public bathrooms if available, do not provide soaps and sanitizers as they are not considered an essential commodity. Therefore, it will be challenging to adequately implement the frequent hygienic hand washing and sanitation of environmental surfaces. Social distancing and isolation of suspected/confirmed COVID-19 cases is essential to stop virus transmission and spread. However, the facilities required to effectively isolate infected individuals or individuals suspected to have been exposed to the virus are non-existent in sub-Sahara Africa. For example, most infected individuals or those exposed to the virus will not have the luxury of a separate bedroom with separate sanitary facilities in their homes. It is common to find 8 people living in a small two bed-roomed house, therefore it will be impossible to institute social distancing and isolation practices in the event of suspected exposure or infection. As part of social distancing, shelter-in or lock- down procedures, limitations have been placed on our movements. We can go to the grocery store to purchase food and other essential products. However, there are limits to how many grocery trips we can make and this means that families will have to buy food in bulk and store it at home. Most of the food we eat such as fresh vegetables, fruits and meats are perishable and thus require refrigeration in order to extend their shelf life. Most people in sub-Sahara Africa do not have refrigerators and even those who have refrigerators are subjected to frequent power interruption and can only buy enough supplies for a few days. This means that frequent visits to the shops and open markets is unavoidable. Food supply, har- vesting etc in most countries will be challenged as lockdowns are implemented to slowdown social interaction and consequently disease spread. We may start to see disruptions in the supply chain, a dip in demand for commodities, loss of income as a result of layoffs and cash flow issues. We have already observed the impact to the food and commodity distribution system. In the most impacted developed countries such as the USA, it has not been uncommon since early March to find empty shelves where commodities such as sanitizers, disinfectants, soaps, toilet paper, paper towels, rice and canned goods used to be found. As a result of the early panic, the demand of these goods exceeded the supply and shortages resulted. Such scenarios would likely become the norm in sub-Sahara Africa. Developed countries such as Canada, Australia, Germany and the UK are paying their citizens to stay at home. The state of our economies in sub-Sahara Africa will not allow us to pay salaries and ensure every member of society has adequate supplies. People will have to calculate the risk of sure starvation if they obey the lockdown measures or risk going out and contracting COVID-19 because they have to make a living. The World Health Organization (WHO) recently released guidance on the use of face masks, as a result countries are instituting rules requiring Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 3 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... citizens to wear face masks from cloth material. In sub-Saharan countries, this has brought up ques- tions of affordability of face masks and most of the population is waiting for government handouts of face masks. While it will be challenging to implement face mask wearing overall, it will be even more challenging to ensure face masks are washed prior to reuse and that people understand that face masks are not a 100% effective barrier. Therefore, people need to be educated on how the face masks should be worn and removed to avoid virus transmission and/or cross contamination as cloth masks are being removed. In addition face masks need to be used with all the hygiene and sanitary practices to control this virus. In light of all this, education of the masses is critical in the control of COVID-19 and needs to be a part of the complete COVID-19 mitigation framework. African culture will play a significant role dur- ing this pandemic as is the case for Ebola and cholera epidemics. For example, most relatives of infected individuals will be unlikely to comply fully with the directives of social distancing and isolation. Adherence to precautionary measures in public places such as shops and open markets that are visited by people from different economic, educational and healthy backgrounds will be chal- lenged (markets in Nigeria, Kenya, Tanzania, Bu- rundi etc). Behavioral norms are the most difficult to break. For example, young people are taught to cough into their fisted hands but now it is recommended to use disposable tissues that are a luxury to many or to cough into the elbow rather than hands [33]. The African culture is based on the use of hands, for example to greet, which makes it difficult to adhere to the new recommendation. Though the washing of hands during Ebola and cholera epidemics has been recommended, many people still walk out of the toilet without washing hands. It will be difficult for people during this pandemic to remember to wash their hands and wash them in the proper manner too [33]. Ubuntu (you are, because I am) is one of the cornerstones of African culture where a sense of community is still being practiced to a large extent. Social distancing becomes a challenge as Africans are very social people. For example, if one is unable to till land on their own, one can invite people in the village to help with the tilling whilst they provide beer and food for the helpers. Social gatherings such as funerals and weddings are held in high esteem. Limiting the number of people who can attend to limit social interactions is con- sidered a taboo. In a typical household people go to different jobs and engage in different activities throughout the day and the question that comes uppermost is can social distancing be implemented in such situations? Sub-Sahara African governments do not have the capacity to construct overflow hospital bed capacity. This study will address the following objectives: First, to determine the total number of individuals likely to be infected with COVID-19. Secondly, to find possible hospitalization strategies that would not overload hospital bed capacity and the number of infected individuals who would need to be safely isolated. Finally, to find a strategy which yields the lowest number of deaths. II. DO PEOPLE LEARN FROM PREVIOUS EPIDEMICS: EBOLA To date Africa has registered very few cases of Covid-19 infections and deaths, even though the cases are rising. One wonders whether frequent exposure of the African population to epidemics has prepared it to prevention protocols. We give examples from Ebola in the Democratic Republic of Congo (DRC) and the Sudan, where prior expo- sure to the epidemic reduced the disease caseload. The data for various epidemics in the DRC and the Sudan presented in Figures 1(a) and 1(b) shows a very high peak of total infections for the first epidemic and declining number of total infected during subsequent epidemics. The data suggests that the populations in the two countries have learnt how to manage an epidemic (Ebola) which requires high standards of hygiene and social distancing. Levy et al, [15] have shown that the severity of an outbreak is linked to the level Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 4 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... (a) 1976 1979 2004 Year 0 50 100 150 200 250 300 N u m b e r o f E b o la C a s e s (b) 1976 1995 2007 2008 2012 2014 Year 0 50 100 150 200 250 300 350 N u m b e r o f E b o la C a s e s Fig. 1: Ebola outbreaks plots for (a) Sudan and (b) the Democratic Republic of Congo. 0 5 10 15 20 25 30 35 40 45 50 Week 0 20 40 60 80 100 120 140 N u m b e r o f N e w E b o la C a s e s (a) 0 5 10 15 20 25 30 35 40 45 50 Week 0 10 20 30 40 50 60 70 80 90 100 N u m b e r o f N e w E b o la D e a th s (b) Fig. 2: Democratic Republic of Congo plots for (a) new Ebola cases (b) new Ebola deaths. of prior knowledge and education of the general population as well as preparedness of health care facilities. Using data from the Democratic Republic of Congo (DRC) for new Ebola cases and new Ebola deaths, we notice a trend of rising to a peak and then declining, indicating two phases (i) naivety to the virus early in the infection and (ii) experience towards adherence to prevention protocols (Figures 2(a) and 2(b)). Ebola and COVID-19 management are simi- lar in many ways. They both rely on very high standards of hygiene, avoiding hand shaking and low density occupancy in residential homes. The difficulty one encounters when modeling Ebola or COVID-19 is modeling the infections arising from poor hygiene, social distancing etc. From Figures 1 and 2, we suggest modeling infections from poor hygiene and lack of social distancing by a function fitted to the data and depicted in Figure 3. This function represents high infections due to poor understanding of the prevention mea- sures early into the epidemic, but as the infection progresses individuals who apply the knowledge from previous epidemics adapt to avoid infections as was the case for Ebola. The function in Figure 3 mimics the process observed for Ebola and is adopted in this study. III. QUALITY OF THE LOCKDOWN Since SARS-2 is a respiratory disease, we want to incorporate the effects of household transmis- sion due to an infected individual i, in household Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 5 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... 0 200 400 600 800 1000 1200 Time (days) 0 5 10 15 20 25 30 35 40 45 50 E n v ir o n m e n t, ( 1 = 0 .6 5 , 2 = 0 .3 3 ) Fig. 3: Time course for the environment h at time t during the lockdown. The rate of exposure from a member of the family j to i is given by: νh = ηg × ψH × ψi,inf (1) where ψH depends on household size and ψi,inf represents asymptomatic or symptomatic status of the infection. An individual in the household h is also exposed to community transmission because a member of the household has to leave the house to transact with the community, for example to buy food or to go to work (if they are front-line workers) during the lockdown. This movement is inevitable in poverty stricken Africa where food insecurity is common and food must be sourced daily. The rate of community exposure is given by: Ch = εg × fg(t) × ψi,age age group, (2) where εg is the baseline exposure from the com- munity, fg is a time dependent curve that modifies the community rate of exposure over time and ψi,age accounts for disease susceptibility depend- ing on age. The rate of exposure of individual i in house- hold h in which a member goes out to transact with the community is given by: λ = Si,g(t)  Mi,h(t) ∑ j 6=i νh + Ch   (3) (3) has been used to moderate transmission of res- piratory synctial virus which is similar to SARS- 2, [13]. The approach described in (1), (2) and (3) is described in detail in [13]. We have used the data from [13] to incorporate the quality of the lockdown and to explain why the number of COVID-19 cases exploded after the lockdown. IV. MODEL DESCRIPTION We develop a simple model in the context of the sub-Sahara Africa environment which consists of a class of individuals, S, who are susceptible to the disease, a class of individuals, E, who have been exposed to the disease, a class of asymptomatic individuals, Ias. These are individuals who are not showing symptoms but are transmitting the virus. A class of symptomatic infectives, Is, a class of individuals who require hospitalization or to be isolated, H, and a class of recovered individuals, R. We note that Ias and Is can infect S directly and indirectly by contaminating the environment φ. For simplicity of notation, let x(t) = (x1(t),x2(t),x3(t),x4(t),x5(t),x6(t)), = (S(t),E(t),Ias(t),Is(t),H(t),R(t)), x7(t) = φ(t), N(t) =x1(t)+x2(t)+x3(t)+x4(t)+x5(t)+x6(t). We consider the following model: dx1 dt = π − (β1x3 + β2x4)x1 N −β3x7x1 −µx1, (4) dx2 dt = (β1x3 + β2x4)x1 N + β3x7x1 −(α1 + µ)x2, (5) dx3 dt = α1x2 − (γ1 + α2 + µ + κ1)x3, (6) Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 6 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... dx4 dt = α2x3 − (γ2 + δ1 + κ2)x4, (7) dx5 dt = γ1x3 + γ2x4 − (δ2 + κ3)x5, (8) dx6 dt = κ1x3 + κ2x4 + κ3x5 −µx6, (9) where (π, µ) =   (0, 0) during lockdown 6= (0, 0) no lockdown, inter-zonal movement allowed. (10) For short lockdown periods one can assume that µ = 0. This assumption is not valid for long lockdowns. The quality of the environment is described by the equation: dx7 dt = α3(x3 + x4) N −µ1x7. (11) The parameter α3 measures the rate of con- taminating the environment, that is, the rate at which the amount of pathogens are released into the environment by both asymptomatic and symp- tomatic infectives. Equation (11) is also supported by Berge et al. [2] for their model on Ebola. The model flow diagram is given below: Fig. 4: Model flow diagram where θ = (β1x3+β2x4)x1 N + β3x7x1. Equation (4) describes the rate of change in the susceptible population, x1. The first term in (4) represents recruitment of people into the sus- ceptible class through movement of people from different zones when this is allowed. This term is zero during a lockdown but may be nonzero if inter-zone movements are allowed. The second term represents loss due to infection of susceptible individuals by asymptomatic infectives, x3, at the rate β1 and symptomatic infectives, x4, at the rate β2. The third term represents indirect infections due to an unclean environment at the rate β3. As time has progressed, it has become necessary to account for loss due to other causes such as natural death at the rate µ. There has been confusion over the number of covid deaths and it has become pathologically necessary to ascertain that a covid infected individual actually died of covid complications, [22], [30]. However, µ was ignored during early stages of the pandemic and every covid infected individual was assumed to have died of covid. Equation (5) describes the rate of change of the ex- posed class, x2. The first two terms represent gain from infection of susceptible individuals and the third term represents loss due to sero-conversion to the asymptomatic state at the rate α1 and natural death at a constant rate µ, respectively. The same comment regarding µ in equation (4) applies in this case. Equation (6) describes the rate of change of the asymptomatic infected class, x3. The first term represents gain from exposed individuals who are converting to sero-positive status without exhibit- ing symptoms. The second term represents loss due to conversion to symptomatic state and hospi- talization at the rates α2 and γ1, respectively. κ1 represents losses from this class due to recovery and µ represents natural death as explained in (10). The first term on the right hand side of equation (7) represents gain from the asymptomatic state. The second term represents loss due to hospital- ization or isolation and a blanket term representing loss due to both natural death and disease induced death at the rates γ2 and δ1 = (µ + δx4 ), re- spectively. Loss from this class due to recovery is assumed to occur at a constant rate κ2. Equation (8) describes the rate of change of the hospitalized and isolated class. The first two terms represent gain from testing and contact tracing of Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 7 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... both asymptomatic and symptomatic individuals. The third term represents loss due to blanket death at the rate δ2 = (µ + δx5 ) and recovery at the rate κ3. Equation (9) describes the rate of change of the re- covered individuals. The first three terms represent gain from recovery of asymptomatic, symptomatic and hospitalized cases. The recovered class loses people through natural death at a rate µ for a long pandemic. Equation (11) describes the rate of contaminating the environment due to the release of pathogens into the environment by symptomatic and asymp- tomatic infectives at the rate α3. The second term represents cleaning of the environment (naturally or due to interventions) at the rate µ1. V. MODEL ANALYSIS A. Positivity of solutions Denote by <6+ the points x(t) = (x1(t), x2(t), x3(t), x4(t), x5(t), x6(t)) in <6 with positive coordinates and consider the system (4) − (9) with initial values x0 = (x01, x 0 2, x 0 3, x 0 4, x 0 5, x 0 6) ∈ < 6 +. We can state the following Lemma: Lemma 1. If x0i ≥ 0, i = 1, ..., 6 then xi(t) ≥ 0 for t > 0, i = 1, ..., 6. Proof: First, we want to show that 0 ≤ x7(t) ≤ α3µ1 where the lower bound signifies a clean environ- ment and the upper bound signifies a Covid-19 contaminated environment. From (11), we have dx7 dt = α3(x3 + x4) N −µ1x7 ≤ α3 −µ1x7. The solution for a totally contaminated COVID-19 environment is given by x7(t) ≤ x7(0) e−µ1 t + α3 µ1 (1 −e−µ1 t). As t → ∞, x7(t) ≤ α3 µ1 . For a clean environment, (11) becomes dx7 dt = α3 (x3 + x4) N −µ1x7 ≥ −µ1x7 x7(t) = x0 e −µ1t ≥ 0. Hence, we have 0 ≤ x7(t) ≤ α3 µ1 . From equation (4), we have dx1 dt = π − (β1x3 + β2x4)x1 N −β3x7x1 −µx1 ≥ π − ( β1 + β2 + β3 α3 µ1 ) x1 ≥ −wx1, w = ( β1 + β2 + β3 α3 µ1 ) . The solution is x1(t) ≥ x01 e −wt ≥ 0, ∀t ≥ 0. Similarly, we can show that for i = 2, ..., 6 xi(t) ≥ 0, i = 2, ..., 6. This completes the proof. B. Invariance The total population, N(t), at time t is given by dN dt = π −µN −δx4x4 −δx5x5 ≤ π −µN. By Gronwall inequality, it is easy to show that 0 ≤ N(t) ≤ π µ . For the existence of a unique bounded solution, we infer that any solution of the system (4) − (11) is non negative and bounded in Ω ={ (x1,x2,x3,x4,x5,x6) ∈<6+ : N ≤ π µ , x7 ≤ α3µ1 } . Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 8 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... C. Disease Free Equilibrium (DFE) 1) Disease Free Equilibrium for π = 0: During a strict lockdown, we can take π = 0. The system (4) − (11) has two equilib- rium points, the disease eradication point ζ0 = (0, 0, 0, 0, 0, 0, 0) and the disease endemic point ζ1 = (x ∗ 1,x ∗ 2,x ∗ 3,x ∗ 4,x ∗ 5,x ∗ 6,x ∗ 7). To establish the stability of ζ0, we use the Hartmann-Grobmann theorem which roughly states that near an equilib- rium point, the dynamics of the original (nonlin- ear) system are the same as those for the linearized system. The linearized system for (4) − (11) is given by: dy(t) dt = Ay(t), where y(t) = (x1(t),x2(t),x3(t),x4(t),x5(t),x6(t),x7(t)) T, A =   0 0 0 0 0 0 0 0 −α1 0 0 0 0 0 0 α1 −b11 0 0 0 0 0 α2 −c11 0 0 0 0 0 γ1 γ2 −d11 0 0 0 0 κ1 κ2 κ3 0 0 0 0 α3 α3 0 0 −µ1   and b11 = (γ1 + α2 + κ1) c11 = (γ2 + κ2 + δ1) d11 = (δ2 + κ3) The eigenvalues of |A−λI| are given by (0, 0,−α1,−b11,−c11,−d11,−µ1). The reproduc- tion number R0, is given by the largest spectral radius of the matrix |A−λI| . In this instance the possible values of R0 are max{α1, b11, c11, d11, µ1}. Since λi < 0, i = 1, 2, ..., 7, the system (4)−(11) is stable and tends to ζ0 as t →∞. Provided the population maintains the covid control protocols and keeps R0 below 1, the disease will fail to establish in the population. 2) Disease Free Equilibrium for π 6= 0: When inter-zone movements are allowed, that is, π 6= 0, the system (4)−(11) has a disease free equilibrium point ζ0 = (πµ, 0, 0, 0, 0, 0, 0). We use the technique by Van den Driessche and Watmough [29] to find the model reproduction number. The matrix for new infections is given by F whilst the matrix for other transitions is given by V, where F =   (β1 x3 +β2 x4)x1 N + β3 x7 x1 0 0 0   , V =   ax2 bx3 −α1 x2 cx4 −α2 x3 µ1 x7 − α3 (x3+x4) N   and a = (α + µ) b = (γ1 + α2 + κ1 + µ) c = (γ2 + κ2 + δ1). The Jacobean matrices F of F and V of V are computed at the point ζ0 with respect to the infected classes (x2, x3, x4, x7). Thus, the basic reproduction number R0, given by the spectral radius of the matrix FV−1 is the maximum of the moduli of the eigenvalues of that matrix FV−1 given by, R0 = Rx3 + Rx4 + Rx7, where, Rx3 = α1 β1 ab Rx4 = α1 α2 β2 abc Rx7 = α1 α3 β3 (c + α2) abcµ1 . Based on R0 we can state the following theorem: Theorem 2. The DFE point, ζ0, is locally asymp- totically stable if R0 < 1 and unstable if R0 > 1. Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 9 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... Remark 1: R0 can be greater than 1 even if Rx3 < 1, Rx4 < 1, Rx7 < 1. It is necessary for each sub-reproduction number Rx3 , Rx4 and Rx7 to be less than 1 and their sum to be less than 1 for the disease to fail to establish in the population. Remark 2: Unlike the case π = 0, now the adher- ence to the covid protocols by each infected sub- population, xi, i = 3, 4, is more strict (adherence to social distancing, mask wearing etc) and the hygiene measures must be more strictly observed for the stability of ζ0 to be achieved. 3) Global stability of the Disease Free Equilib- rium: Theorem 3. The DFE point, ζ0, of system (4) − (11) is globally asymptotically stable for R0 < 1. Proof. To prove theorem 3, we use Kamgang- Sallet stability theorem, [10]. Let Z = (Z1, X2) with Z1 = (x1, x6) ∈ <2 and Z2 = (x2, x3, x4, x5, x7) ∈ <5. In terms of Z1 and Z2 system (4) − (11) can be written as: Ż1 = A1 (Z)(Z1 −Z∗1 ) + A12(Z) Z2 Ż2 = A2(Z) (Z2) where Z∗1 = ( π µ , 0), with A1(Z) = [ −µ 0 0 −µ ] , A12(Z) = [ 0 −β1x1 N −β2x1 N 0 −β3x1 0 κ1 κ2 κ3 0 ] and A2(Z) =   −a β1x1 N β2x1 N 0 β3x1 α1 −b 0 0 0 0 α2 −c 0 0 0 γ1 γ2 −d 0 0 α3 N α3 N 0 −µ1   , where d = (δ2+κ3). We want to show that the five sufficient conditions of Kamgang-Sallet Theorem in [10] are satisfied as follows: (i) The system (4)−(11) is a dynamical system on Ω, as defined and shown in section V . (ii) The eigenvalues of A1(Z) are real and nega- tive, thus the system Ż1 = A1(Z) (Z1−Z∗1 )+ A12(Z) Z2 is globally asymptotically stable at the equilibrium Z∗1 . (iii) The matrix A2(Z) is a Metzler matrix, i.e. a matrix such that off diagonal terms are non negative and is irreducible for any given Z ∈ Ω. (iv) There exists a matrix Ā2, which is an upper bound for the set M = A2(Z) : Z ∈ Ω. Indeed, Ā2 =   −a β1 β2 0 β3x∗1 α1 −b 0 0 0 0 α2 −c 0 0 0 γ1 γ2 −d 0 0 α3 N∗ α3 N∗ 0 −µ1   is an upper bound for M. (v) For R0 < 1 , λ is the eigenvalue of Ā2, α(Ā2) = max{Re(λ) : λ}≤ 0 To check condition (v) we will use the following lemma which is a characterization of Metzler stable matrices: Lemma 4. Let M be a square matrix written in block form D = [ A B C D ] with A and D being square matrices. M is Metzler stable if and only if matrices A and D −CA−1B are Metzler stable. Using Lemma 4, matrix Ā2 can be expressed in the form of matrix M with: A =  −a β1 β2α1 −b 0 0 α2 −c   , B =  0 β3x∗10 0 0 0   C = [ 0 γ1 γ2 0 α3 N∗ α3 N∗ ] , D = [ −d 0 0 −µ1 ] . Clearly A, is a stable Metzler matrix and after computations we obtain D − CA−1B is a stable Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 10 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... Metzler matrix if and only if Rmax0 = α1 β1 ab + α1 α2 β2 abc + α1 α3 β3 (c + α2) abcµ1 ≤ 1. D. Endemic Equilibrium Point (EEP) Solving for the system of equations (4) − (11) by equating the RHS to zero, we find the coordinates of the EEP given by ζ1 = (x∗1, x ∗ 2, x ∗ 3, x ∗ 4, x ∗ 5, x ∗ 6, x ∗ 7), where x∗1 = abcN∗µ1 α1 a33 (12) x∗2 = a22 (R0 − 1) aα1 a33 (13) x∗3 = a22 (R0 − 1) aba33 (14) x∗4 = α2 a22 (R0 − 1) abca33 (15) x∗5 = a22 a44 (R0 − 1) abcda33 (16) x∗6 = κ3 a22 a44 a55 (R0 − 1) abcdµa33 (17) x∗7 = α3(α2 + c)a22(R0 − 1) abcN∗µ1 a33 (18) and a22 = abcN ∗µµ1 a33 = (c + α2) α3 β3 + (cβ1 + α2 β2) µ1 a44 = (α2 γ2 + cγ1) a55 = (cdκ1 + dκ2 α2). The coordinates (13) − (18) exist if and only if R0 > 1. 1) Global stability of the Endemic Equilibrium Point: The global stability of the EEP is explored by proving the following theorem: Theorem 5. If R0 > 1 then the EEP given by ζ1 is globally asymptotically stable in the region Ω. Proof. Following the work of [5], we construct a Lyapunov function L of the type: L(xi) = 7∑ i=1 ( xi −x∗i −x ∗ i ln xi x∗i ) . Differentiating L with respect to xi gives: dL dt = 7∑ i=1 ( xi −x∗i xi ) dxi dt . Substituting for dxi dt , i = 1, ..., 7, we get dL dt = ( x1 −x∗1 x1 ) [ π − (β1x3 + β2x4)x1 N −β3x7x1 −µx1 ] + ( x2 −x∗2 x2 ) [ (β1x3 + β2x4)x1 N + β3x7x1 −ax2 ] + ( x3 −x∗3 x3 ) [α1x2 − bx3] + ( x4 −x∗4 x4 ) [α2x3 − cx4] + ( x5 −x∗5 x5 ) [γ1x3 + γ2x4 −dx5] + ( x6 −x∗6 x6 ) [κ1x3 + κ2x4 + κ3x5 −µx6] + ( x7 −x∗7 x7 )[ α3(x3 + x4) N −µ1x7 ] = A33 −A22, Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 11 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... where A22 = ( x1−x∗1 x1 )2[ (β1x3 +β2x4) N +β3x7 +µ ] + [ x∗1 x1 π+ (β1x3 +β2x4)x ∗ 1 N +β3x7x ∗ 1 +µx ∗ 1 ] + ( x2 −x∗2 x2 )2 a + ax∗2 + x∗2 x2 [ β1x3 + β2x4)x1 N + β3x7x1 ] + ( x3 −x∗3 x3 )2 b + bx∗3 + x∗3 x3 α1x2 + ( x4 −x∗4 x4 )2 c + cx∗4 + x∗4 x4 α2x3 + ( x5−x∗5 x5 )2 d+dx∗5 + x∗5 x5 (γ1x3 +γ2x4) + ( x6 −x∗6 x6 )2 µ + µx∗6 + x∗6 x6 (κ1x3 + κ2x4 + κ3x5) + ( x7−x∗7 x7 )2 µ1 +µ1x ∗ 7 + x∗7 x7 α3(x3 +x4) N and A33 = π + x∗ 2 1 x1 [ (β1x3 + β2x4) N + β3x7 + µ ] + [ (β1x3 + β2x4) N + β3x7 + µ ] + x∗ 2 2 x2 a + α1x2 + x∗ 2 3 x3 b + α2x3 + x∗ 2 4 x4 c +γ1x3 + γ2x4 + x∗ 2 5 x5 d + κ1x3 + κ2x4 +κ3x5 + x∗ 2 6 x6 µ+ α3(x3 + x4) N + x∗ 2 7 x7 µ1. Since all the parameters used in the system (4) − (11) are non negative we have dL dt ≤ 0 for A33 ≤ A22 and dLdt = 0 if and only if x1 = x ∗ 1,x2 = x∗2,x3 = x ∗ 3,x4 = x ∗ 4,x5 = x ∗ 5,x6 = x ∗ 6,x7 = x ∗ 7. Thus by La-Salle’s invariance principle, the EEP is globally asymptotically stable. TABLE I: Numerical values for the parameters of the Italian case Parameter Value/range Source β1 0.492 [8] β2 1.30 [20] β3 0 estimate δ1 0.015 [19] δ2 4827 53578 [33] α1 1 3.21 [20] α2 1 2.27 [20] α3 0 estimate γ1 [0, 0.6] estimate γ2 [0, 0.6] estimate κ1 1 6 [8] κ2 1 10 [8] κ3 1 14 [8] µ 0.00003032 [35] µ1 0 estimate π µ× 400 000 [11] VI. NUMERICAL SIMULATIONS We want to find optimal isolation rates γ1 and γ2, using Matlab programs, under which the number of infected individuals will not overwhelm the hospital capacity, H. We performed numerical simulations using data from the Italian coronavirus epidemic from 31st January to the 15th of May 2020 to demonstrate that our model can accurately reproduce the recorded data on numbers of in- fected and dead individuals. The parameter values used for numerical simulations are given in Table I. 1) Sensitivity Analysis: Five parameters, β3, α3, γ1, γ2 and µ1 in Table I have been estimated based on the number of hospitalized cases in the Lombardy region of Northern Italy. We have analyzed how sensitive the reproduction number is to the changes in these five parameters (Figure Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 12 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... PRCC P a r a m e t e r Fig. 5: A diagram showing the sensitivity of R0 to various model parameters. 5). The parameters α3 and β3 are important and provide insight into understanding how the sub- Sahara Africa situation will differ from the Italian or in general the situation in Western countries, where we assumed α3 = 0. A. Italy Lombardy is a region with a population of ap- proximately 10 million. By an iterative procedure, since we know the number of people infected with the coronavirus from 31st January to 22nd March 2020, we have estimated that the number of people susceptible to infection by the virus, through their failure to observe prevention measures such as self quarantine, social distancing etc at the start of the coronavirus epidemic, was about 400 000. As at 31st January 2020, the number who were potentially exposed to the infection is estimated to be around 200 individuals. On the 31st January 2020, Italy recorded its first 3 cases of individuals infected with the coronavirus. Based on this initial data, we present examples of strategies by fixing the rate of isolating symptomatic infectives and then finding the corresponding rate of isolating asymptomatic infectives which ensures that the combined number of infectives does not exceed the hospital bed capacity and vice versa. The examples discussed here are not unique but are typical of other scenarios and provide insight into the following: (i) how the infection curve can be flattened to ensure that the hospital bed capacity is not exceeded. (ii) For non optimal cases, where hospital bed capacity is exceeded, to quantify the number of infectives for each non optimal case which must be safely isolated outside hospital facil- ities. According to [23], Italy has 12.5 beds per 100 000 individuals of Intensive Care Unit (ICU) or critical care beds (CCB) beds and 3.18 beds per 1 000 individuals ordinary hospital beds. For Lombardy, this data (combining both ordinary and ICU hospital beds) gives 33 182 beds. Figure 6 presents an example for a fixed rate γ2 = 0.33 of isolating symptomatic infectives. We find that the optimal rate of securing asymptomatic infectives when there is a lockdown, π = 0, should be γ1 = 0.46. Converting these rates to time, we see that asymptomatic infectives should be identified, hospitalized in a time almost 1.5 times faster than the time of hospitalizing symp- tomatic infectives. Currently, every nation is ex- periencing a lack of materials and equipment to conduct tests. This rate of hospitalizing asymp- tomatic infectives would be difficult to achieve as there has been a shortage of testing materials, and most governments have decided they would only test individuals who present with COVID-19 symptoms. Moreover, the time it is taking to obtain test results 2−3 days is slowing down the testing significantly. For this strategy, we have determined the num- ber of infected people that should be safely secured for varying values of γ1 at peak infection. For γ1 = 0, the number in excess of hospital bed ca- pacity that must be secured is 22 408, for γ1 = 0.3, the number in excess of hospital bed capacity that must be secured is 13 407. For the non optimal cases, if the hospital overflow bed capacity is increased by 50% (as was done in most Western countries) the only case that would have accommo- Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 13 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... 0 50 100 150 200 250 Time (days) 0 1 2 3 4 5 6 H o s p it a li s e d , ( 2 = 0 .3 3 ) 10 4 Hospital bed capacity=33182 1 =0, ~=0 1 =0.3 1 =0.49 1 =0, =0 1 =0.3 1 =0.46 data1 Fig. 6: Population level plots for hospitalized in- dividuals for varying values of γ1 dated the overflow is (γ1,γ2) = (0.3, 0.33). Figure 6 shows that when inter zonal movement is al- lowed, π 6= 0, the number of people needing hos- pitalization or to be safely isolated increases. The optimum removal rates with inter-zonal movement are given by (γ∗1,γ ∗ 2 ) = (0.49, 0.33), implying that asymptomatic infectives must be isolated in a time 1.6 times faster than when there is no inter zonal movement. The optimal solutions in Figure 6 show that the peak hospitalization capacity occurs earlier when there is a lockdown (π = 0) than when there is inter-zonal movement (π 6= 0). Figure 7 presents a strategy where the rate of hospitalizing symptomatic infectives is fixed at γ2 = 0.6. The optimal rate for γ1 which flattens the hospital admission curve is found to be γ1 = 0.16. In other words, almost one third of the effort must be devoted to testing, contact tracing and hospitalizing asymptomatic infectives. For this strategy the situation regarding the non optimal cases is as follows: for γ1 = 0 the number needed to be secured is 10 097, which is too high to accommodate. Using optimization techniques to find γ∗1 and γ ∗ 2 we have found and demonstrated in Figure 8 that the optimum rates are (γ∗1,γ ∗ 2 ) = (0.6, 0.26). This optimum pair implies that the effort devoted to 0 50 100 150 200 250 Time (days) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 H o s p it a li s e d , ( 2 = 0 .6 ) 10 4 Hospital bed capacity=33182 1 =0 1 =0.16 data1 Fig. 7: Population level plots for hospitalized in- dividuals for varying values of γ1 0 50 100 150 200 250 Time (days) 0 1 2 3 4 5 6 H o s p it a li s e d , ( 1 = 0 .6 ) 10 4 Hospital bed capacity=33182 2 =0.1 2 =0.26 data1 Fig. 8: Population level plots for hospitalized in- dividuals for varying values of γ2 testing, contact tracing and isolating symptomatic infectives must be 2.3 times higher than that of isolating asymptomatic infectives. Figure 9 shows that for γ1 = 0.3 the optimum pair is (γ∗1,γ ∗ 2 ) = (0.3, 0.45). This strategy targets to remove symptomatic infectives faster than the asymptomatic infectives. This strategy removes symptomatic infectives in a time at least one and half times faster than the time of isolating Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 14 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... TABLE II: Comparison of deaths at different optimal cases Strategy γ1 γ2 30 days 60 days 90 days 105 days 105 days δ = 0.01 δ = 0.03 δ = 0.02 δ = 0.015 δ = 0.09 Lockdown No lockdown 1 0.49 0.33 210 10 126 32 108 31 433 120 200 (29) (11 591) (27 682) (31 368) 2 0.16 0.6 226 12 861 33 994 31 202 115 690 (29) (11 591) (27 682) (31 368) 3 0.6 0.26 200 8 989 30 734 31 162 120 640 (29) (11 591) (27 682) (31 368) 4 0.3 0.45 217 11 179 32 771 31 135 117 600 (29) (11 591) (27 682) (31 368) 0 50 100 150 200 250 Time (days) 0 1 2 3 4 5 6 H o s p it a li s e d , ( 1 = 0 .3 ) 10 4 Hospital bed capacity=33182 2 =0.1 2 =0.3 2 =0.45 data1 Fig. 9: Population level plots for hospitalized in- dividuals for varying values of γ2 asymptomatic infectives. To illustrate the impact of non adherence to prevention measures, such as social distancing, not wearing masks etc, on the optimal case given in Figure 9, we considered how the case β3 6= 0 for Italy would have altered the conclusions presented in Figures 6 to 9. Figure 10 shows that if we vary the parameter α3 the hospital bed capacity for Lombardy in Italy would have been exceeded for any α3 ≥ 0.01. This suggests that high stan- dards of hygiene are key to controlling COVID-19 infections. We can see from the examples of the four strategies illustrated in Figures 6, 7, 8, 9 that there is no unique way of flattening the curve in order to 0 50 100 150 200 250 Time (days) 0 2 4 6 8 10 12 14 H o s p it a li s e d , ( 1 = 0 .3 , 2 = 0 .4 5 ) 10 4 Hospital bed capacity=33182 3 =0 3 =0.05 3 =0.1 data1 Fig. 10: Population level plots for hospitalized individuals for varying values of α3 protect the hospital bed capacity. The question we address is how does one choose the best strategy among the optimal strategies? Table II gives the number of deaths resulting from the four strategies above and compares each strategy with the actual number of recorded deaths. We conclude that the best strategy is one which reduces the number of deaths. In this case (Table II) any strategy that isolates the symptomatic infectives faster is preferable as it is more economical. B. Optimal Control 1) Optimal control without incorporating household and community exposure: To study measures that reduce disease transmission, Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 15 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... such as lockdown or isolation of infectives, we introduced two controls, u1 and u2, on the infection coefficients β1 and β2 as shown below: dx1 dt = π−(1−u1) β1x3x1 N −(1−u2) β2x4x1 N −β3x7x1 −µx1 (19) dx2 dt = (1 −u1) β1x3x1 N + (1 −u2) β2x4x1 N +β3x7x1 − (α1 + µ)x2 (20) dx3 dt = α1x2 − (γ1 + α2 + µ + κ1)x3 (21) dx4 dt = α2x3 − (γ2 + δ1 + κ2)x4 (22) dx5 dt = γ1x3 + γ2x4 − (δ2 + κ3)x5 (23) dx6 dt = κ1x3 + κ2x4 + κ3x5 −µx6 (24) and dx7 dt = α3(x3 + x4) N −µ1x7 (25) We want to minimize the objective functional given by J(u1,u2) = ∫ tT 0 A1x2 + A2x3 + A3x4 + 1 2 A4u 2 1 + 1 2 A5u 2 2. (26) The goal is to find a set of controls that minimize the number of susceptible individuals who come into contact with infected individuals. Let u∗1, u ∗ 2 be the optimal controls. The problem is to find J(u∗1,u ∗ 2) = minJ(u1,u2) , (u1,u2) ∈ U, (27) subject to system (19) − (25), where U is the set of measure functions defined from [0, tT ] to [0, 1]. The optimality conditions are given by, u∗1 = min { max [ 0,D1 ( β1x3x1 A4N )] ,u1max } , u∗2 = min { max [ 0,D1 ( β2x4x1 A5N )] ,v2max } , where D1 = (λx2 −λx1 ). Calculation for u∗1 and u ∗ 2 is based on Pontrya- gin’s Maximum Principle (see [14]) for a detailed description. 2) Optimal control incorporating household and community exposure.: To study the impact that household and community exposure has on disease transmission control measures, we modify equations (19) − (25) by adding an additional term λ given in (3) which captures household and community exposure of an individual. The modified equations are given by: dx1 dt = π − (1 −u1 + λ) β1x3x1 N −(1 −u2 + λ) β2x4x1 N −β3x7x1 −µx1 (28) dx2 dt = (1 −u1 + λ) β1x3x1 N +(1 −u2 + λ) β2x4x1 N + β3x7x1 −(α1 + µ)x2 (29) dx3 dt = α1x2 − (γ1 + α2 + µ + κ1)x3 (30) dx4 dt = α2x3 − (γ2 + δ1 + κ2)x4 (31) dx5 dt = γ1x3 + γ2x4 − (δ2 + κ3)x5 (32) dx6 dt = κ1x3 + κ2x4 + κ3x5 −µx6 (33) and dx7 dt = α3(x3 + x4) N −µ1x7 (34) where λ is given by (3). The optimality conditions that we want to sat- isfy are the same as those for equations (19)−(25). We consider the non optimal strategy in Figure 6 using equations (19) − (25) and equations (28) − (34) for γ1 = 0.3 and γ2 = 0.33 where u1 = 0 and u2 = 0. If we use the controls u∗1 ≥ 0.2, u ∗ 2 ≥ 0.2 and (γ1, γ2) = (0.3, 0.33) we obtain Figures 11(a) − 11(e). It is clear from Figure 11(e) that household and community disease transmission would increase the number hospitalized though the numbers would still be below the hospital bed capacity for optimal values of γ1 and γ2. The number of susceptible individuals (Figure 11(a)) would decline due to increased infection rates. Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 16 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... 0 50 100 150 Time (days) 0.5 1 1.5 2 2.5 3 3.5 4 S u sc e p ti b le s 10 5 (a) Without Controls With Controls including household and community exposure With Controls excluding household and community exposure 0 50 100 150 Time (days) 0 0.5 1 1.5 2 2.5 3 3.5 4 E x p o se d 10 4 (b) Without Controls With Controls including household and community exposure With Controls excluding household and community exposure 0 50 100 150 Time (days) 0 2000 4000 6000 8000 10000 12000 14000 A sy m p to m a ti c I n fe c ti v e s (c) Without Controls With Controls including household and community exposure With Controls excluding household and community exposure 0 50 100 150 Time (days) 0 2000 4000 6000 8000 10000 12000 S y m p to m a ti c I n fe c ti v e s (d) Without Controls With Controls including household and community exposure With Controls excluding household and community exposure 0 50 100 150 Time (days) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 H o sp it a li z e d 10 4 (e) Hospital bed capacity=33182 Without Controls With Controls including household and community exposure With Controls excluding household and community exposure data1 Fig. 11: A comparison of state variables with and without controls C. Sub-Sahara Africa: South Africa as an exam- ple. To provide insight into the sub-Sahara Africa outlook, simulations were done using data from South Africa, one of the countries in this region with the highest number of coronavirus cases and with the best medical facilities. It is our view that if South Africa cannot cope then most, if not all, countries in sub-Sahara Africa would not cope. On the 4th of March 2020, South Africa recorded its first case of the coronavirus. As of the 1st of April 2020 the highest case counts of coronavirus had been reported in 3 provinces namely Gauteng, KwaZulu Natal and the Western Cape. The 3 provinces had 1 380 reported coron- avirus cases distributed as follows: 645 in Gauteng, 326 in KwaZulu Natal and 186 in the Western Cape. These 3 provinces have a total population of 33 309 473 people. The number of people infected with the coronavirus from the 4th of March to the 1st April 2020 is known, we have estimated that the number of people susceptible to infection by Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 17 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... TABLE III: Numerical values for the parameters of the South African case Parameter Value/range Source β1 0.492 [8] β2 1.30 [20] β3 0.001006082 [11] δ1 0.015 [19] δ2 5 1380 [33] α1 1 3.21 [20] α2 1 2.27 [20] α3 5.3346 × 10−6 estimate γ1 [0, 0.77] estimate γ2 [0, 0.33] estimate κ1 1 6 [8] κ2 1 10 [8] κ3 1 14 [8] µ 0.00004290 [36] µ1 0.00274 [11] π µ × 1 200 000 [11] the virus through failure to self quarantine, self isolate, observe social distancing etc was about 1 200 000. At that time the number who were potentially exposed to the infection is estimated to be about 1 000 individuals. Except for β3, π, α3, µ, µ1 and δ2, we use the parameters in Table 1 for Italy. This is justified on the basis that Italian family bonds are similar to sub-Sahara Africa. The values used for this simulation are given in Table 3. According to [4], Gauteng province, KwaZulu Natal Province and the Western Cape province combined have a hospital bed capacity of 62 787. This number includes both ICU and ordinary hospital beds. Figure 12 compares real time epidemic curves for Italy, Spain, the United Kingdom and South Africa for the first 40 days of the epidemic. Each country implemented the lockdown strategy at different stages of infection. Italy introduced the lockdown on the 9th of March 2020 when the total number of individuals infected with COVID- 19 was 9 172. Spain introduced the lockdown on the 15th of March 2020 when the total number of those infected was 7 798. The United Kingdom introduced the lockdown on the 23rd of March 2020 when the number of individuals infected was 6 650. South Africa introduced the lockdown on the 27th of March 2020 when the number infected was only 1 170. Figure 12 shows different infection trends for the four countries. It is obvious that South Africa which introduced the lockdown early enough displays an epidemic which rises at a gentle rate. We consider the problem of finding the hospitalization rates γ1 and γ2 for which the hospital bed capacity of 62 787 would suffice. Fig. 12: Total covid cases, UK, Spain, Italy, South Africa 1) Scenario with no lockdown measures: Fig- ure 13 shows that for a fixed rate γ2 = 0.33 of symptomatic infectives, the least rate of isolating asymptomatic infectives should be γ1 = 0.77 for φ = 0 (implying that the population must observe all the prevention measures ,social dis- tancing, mask wearing etc) but that for φ 6= 0 the hospital bed capacity is never sufficient. Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 18 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... 0 50 100 150 200 250 Time (days) 0 0.5 1 1.5 2 2.5 H o s p it a li s e d , ( 2 = 0 .3 3 ) 10 5 Hospital bed capacity=62787 1 =0 1 =0.3 1 =0.6 1 =0.77 data1 Fig. 13: Population level plots for hospitalized individuals for varying values of γ1 For the strategies in Figure 13, the numbers of people that must be isolated are given in Table 4. Table 4 presents cases when the hospital bed ca- pacity, denoted by H, is increased by 50% as was done in Italy, Spain and the State of New York, USA. The overflow is given by (HP −H × 1.5). We can see that no pair of isolating infectives γ1 and γ2 would accommodate the number of infected individuals in the 3 provinces of South Africa within hospital facilities. The overflow ca- pacity would be too large for any non-optimal pair γ1 and γ2 and φ 6= 0, to safely isolate the overflow infected population. The only feasi- ble solution is hospitalizing at the optimal rates (γ∗1,γ ∗ 2 ) = (0.77, 0.33) with φ = 0. This option requires isolating asymptomatic infectives faster than symptomatic infectives, a strategy which re- quires perfect hygiene and sanitary measures. This requirement cannot be met in sub-Sahara Africa given the state of the economies. To illustrate the impact of non adherence to disease prevention measures on the optimal case in Figure 13, we varied the parameter α3. Figure 14 shows how the optimal solution in Figure 13 is altered by varying α3. The hospital bed capacity is exceeded for any case α3 6= 0. For the case φ 6= 0, no matter how small φ is, no optimal strategy exists for the three provinces of South Africa. It is unrealistic to expect perfect adherence to social distancing, wearing of masks and lockdown measures, in a region where the population survives on less than $1 per day and there are no food banks. 0 50 100 150 200 250 Time (days) 0 1 2 3 4 5 6 H o s p it a li s e d , ( 2 = 0 .3 3 ) 10 5 Hospital bed capacity=62787 3 =0 3 =0.05 3 =0.1 3 =0.2 data1 Fig. 14: Population level plots for hospitalized individuals for varying values of α3 2) Effect of early lockdown: South Africa in- troduced the lockdown very early. The strategy discussed in section C.1 is therefore not relevant. Hence, we consider the modified system with control given in (19) − (25) and (28) − (34). We consider hypothetically how the lockdown could have altered a non optimal strategy in Figure 13. We chose γ1 = 0.6 and γ2 = 0.33 (u∗1 ≥ 0.1, u∗2 ≥ 0.1) to illustrate this example (see Figures 15(a) − 15(e)). The number of deaths for a model with controls for South Africa for different optimal cases of γ1 and γ2 is given in Table V . Table V compares the actual number of recorded deaths, the controlled number of deaths and the uncontrolled number of deaths (no lockdown). It is clear that the lockdown was very effective. VII. CONCLUSION The analysis has revealed that there are several strategies that can flatten the infection curve and protect hospital bed capacity. However, we have Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 19 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... TABLE IV: Number of people who must be hospitalized. γ1 γ2 H H × 1.5 H peak (HP ) Oveflow 0 0.33 62 787 94 180 246 174 151 994 0.3 0.33 62 787 94 180 207 654 113 474 0.6 0.33 62 787 94 180 112 952 18 772 0 50 100 150 Time (days) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 E x p o se d 10 4 (b) Without Controls With Controls including household and community exposure With Controls excluding household and community exposure 0 50 100 150 Time (days) 0 2000 4000 6000 8000 10000 12000 A sy m p to m a ti c I n fe c ti v e s (c) Without Controls With Controls including household and community exposure With Controls excluding household and community exposure 0 50 100 150 Time (days) 0 2000 4000 6000 8000 10000 12000 S y m p to m a ti c I n fe c ti v e s (d) Without Controls With Controls including household and community exposure With Controls excluding household and community exposure 0 50 100 150 Time (days) 0 2 4 6 8 10 12 H o sp it a li z e d 10 4 (e) Hospital bed capacity=62787 Without Controls With Controls including household and community exposure With Controls excluding household and community exposure data1 Fig. 15: A comparison of state variables with and without controls found that of all the strategies, the preferred strat- egy is either that which removes asymptomatic infectives much faster than the symptomatic in- fectives or the strategy that removes symptomatic infectives faster than asymptomatic infectives as either of these strategies gives the least number of deaths. The optimal control analysis suggests that if Italy had introduced the lockdown much earlier the number of deaths would have been reduced Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 20 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 M H Machingauta, B Lungu, E M Lungu, COVID-19 changing the face of the world. Can sub-Sahara ... TABLE V: Comparison of deaths at different optimal cases γ1 γ2 30 days 60 days 72 days 90 days δ = 0.003 δ = 0.01 δ = 0.01 δ = 0.01 0.77 0.33 no lockdown 178 3 358 5 930 12 394 lockdown 7 151 211 327 actual (5) (123) (238) (705) 0.4 0.6 no lockdown 187 3 805 6 834 14 287 lockdown 7 153 216 339 actual (5) (123) (238) (705) significantly and the hospital bed capacity would have been protected. South Africa introduced the lockdown early. This is evident from the flatness of the infection curve (Figure 12). It is obvious from Fig.12 that early lockdown slowed down the number of infec- tions. The question is why despite this measure the number of infected individuals has kept on rising. The number of infections in fact rose rapidly after the lockdown. It is clear from Figure 15 that the lockdown was ineffective because the individuals in various households came out to mingle with individuals from other households at community places such as shops, markets etc. The lockdown did not have the desired effect. From Figure 15 it is evident that if household and community transmission had been avoided due to higher levels of adherence to control measures the situation would have resulted in 33% fewer asymptomatic infectives, 33% fewer symptomatic infectives and 62% less hospitalizations. The difficulty for South Africa, and indeed any sub-Sahara African country, will be how to ensure the populations living in high density areas observe social distancing and good hygiene practices. 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Pneu- monia in Wuhan, China JAMA Intern Med. 2020 Biomath 10 (2021), 2103117, http://dx.doi.org/10.11145/j.biomath.2021.03.117 Page 23 of 23 http://dx.doi.org/10.11145/j.biomath.2021.03.117 Introduction Do people learn from previous epidemics: Ebola Quality of the Lockdown Model Description Model Analysis Positivity of solutions Invariance Disease Free Equilibrium (DFE) Disease Free Equilibrium for =0 Disease Free Equilibrium for =0 Global stability of the Disease Free Equilibrium Endemic Equilibrium Point (EEP) Global stability of the Endemic Equilibrium Point Numerical Simulations Sensitivity Analysis Italy Optimal Control Optimal control without incorporating household and community exposure Optimal control incorporating household and community exposure. Sub-Sahara Africa: South Africa as an example. Scenario with no lockdown measures Effect of early lockdown Conclusion References